1. Field of the Invention
The present invention relates to a shape data processing method and, more particularly, to a method used by a CAD/CAM (computed aided design/computer aided manufacturing) system in measuring the volume of a product using the shape data thereof.
2. Description of the Prior Art
Heretofore, a number of methods for creating shape data of an object having sculptured surfaces have been proposed (including U.S. Pat. No. 4,789,931 and 4,819,192). These methods create the data by expressing surfaces using vector functions comprising a cubic Bezier equation and by connecting such surfaces under the condition of tangent plane continuation.
As shown in FIG. 19, a designer illustratively designates node vectors P(00), P(03), P(33)1, P(30)l, P(33)2 and P(30)2 in a three-dimensional space. Two spaces are formed using these node vectors, one enclosed by four contiguous node vectors P(00), P(03), P(33)1 and P(30)l, the other by P(00), P(03), P(33)2 and P(30)2. A surface (called a patch) made of these spaces is defined by the following cubic Bezier equation: EQU S(u,v)=(1-u+u E).sup.3 (1-v.div.vf).sup.3 .multidot.P(00) 1)
which provides a vector S ( u , v ) , where u and v are parameters representing the u and v directions respectively. To the node vector P( 00 ) made of control points , the following equations using shift operators E and F apply: EQU E.multidot.P(i,j)=P(i+1, j)(i,j=0,1,2) (2) EQU F.multidot.P(i,j)=P(i,j+1) (i,j=0,1,2) (3) EQU 0.ltoreq.u.ltoreq.1 (4) EQU 0.ltoreq.v.ltoreq.1 (5)
Then control point vectors P(01), P(02), P(10)l, P(20)l to P(23)1, P(31)l and P(32)l are set for the space enclosed by the four node vectors P(00), P(03),P(33)l and P(30)l; vectors P(01), P(02), P(10)2 to P(13)2, P(20)2 to P(23)2, P(31)2 and P(32)2 are set for the space enclosed by the vectors P(00), P(03), P(33)2 and P(30)2. This allows patch vectors S(u, v)l and S(u, v)2 to be generated for the surface shape determined by the control vectors P(01) through P(32)1 and P(01) through P(32)2 via passage through the four node vectors P(00), P(03), P(33)1 and P(30)l as well as through the vectors P(00), P(03), P(33)2 and P(30)2.
Furthermore, at the patch vectors S(u, v)I and S(u, v)2, internal control point vectors P(11)l and P(12)l as well as P(11)2 and P(12)2, with common control point vectors P(01) and P(02) sandwiched therebetween, are modified and set again. This allows the patch vectors S(u, v)l and S(u, v)2 to be connected smoothly.
Specifically, the internal control point vectors P(11)l and P(12)l as well as P(11)2 and P(12)2 are modified and set again as follows. Using node vectors P(00), P(30)l, P(33)1, P(03), P(33)2 and P(30)2 given by framing, control side vectors al and a2 as well as cl and c2 are set to meet the condition of tangent plane continuation on a boundary curve COM12 of contiguous patch vectors S(u, v)l and S(u, v)2. These control side vectors are used to modify and set again the internal control point vectors P(11)l and P(12)l as well as P(11)2 and P(12)2.
When the above method is applied to the other boundary curves surrounding the patch vectors S(u, v)l and S(u, v)2, these patch vectors are eventually connected smoothly under the condition of tangent plane continuation with contiguous patches.
In the description above, a tangent plane is a plane formed by tangent vectors in the tl and v directions at various points of the boundary curves. For example, the condition of tangent plane continuation is met when the tangent plane is the same for the patch vectors S(u, v)l and S(u, v)2 at each of the points on the boundary curve COM12 in FIG. 19.
The condition of tangent plane continuation regarding a point (0, v) on the boundary curve COM12 is determined as depicted in FIG. 20. That is, for the patch vector S(u, v)l, the equation EQU n1=Ha.times.Hb (6)
represents a normal vector nl regarding a tangent vector Ha traversing the boundary curve COM12 (i.e., direction) as well as regarding a tangent vector Hb along the boundary curve COH12 (v direction). For the patch vector S(u, v)2, the equation EQU n2=Hc.times.Hb (7)
represents a normal vector n2 regarding a tangent vector Hc traversing the boundary curve COM12 as well as regarding the tangent vector Hb along the boundary curve COM12.
To meet the condition of tangent plane continuation Under the above constraints requires that the tangent vectors Ha and Hb as well as Hc and Hb exist on the same plane. As a result, the normal vectors nl and n2 are oriented in the same direction.
To satisfy these conditions requires setting the internal control point vectors P(11)l through P(22)1 and P(11)2 through P(22)2 in such a manner that the equation ##EQU1##
will hold, where .lambda.(v), .mu.(v) and .nu.(v) are scalar quantities,
As described, where the designer wants to design a certain external shape, parametric vector functions are used to represent the shape through the use of the boundary curves surrounding each framing space involved.
With the external shape thus generated, a metal mold is prepared and products are manufactured therewith. In advance of actual manufacture, it is convenient to know the volume and other data about the product. Illustratively, knowing the product volume affords the benefit of being able to determine how much raw material is needed to produce a given amount of products.
The typical prior art method for obtaining volumes from external shapes has one notable disadvantage. That is, the shape represented by sculptured surfaces tends to be very complex and changes in such a subtle manner that it often defies precise measurement by conventional procedures which are relatively crude in methodology. With some shapes, there is no conventionally feasible way of measuring the volume thereof.
Consider for example a hollow object shown in FIG. 21. Conventional methods are incapable of distinguishing the hollow portion from the actual object portion. It is virtually impossible to automatically measure the volume of such objects.
Then consider variations of the object illustrated in FIG. 22, the object having its top and bottom left open. FIG. 23 shows one variation whose side wall is discontinued. It is difficult conventionally to measure the volume of this variation of the object.
FIG. 24 shows an object formed with its parts overlapping with one another. This type of object occurs when pipes are connected in a complicated manner. These objects are also difficult to measure in terms of volumes.